A canonical Ramsey theorem for even cycles in random graphs

The celebrated canonical Ramsey theorem of Erdős and Rado implies that for 2 less than or equals k element of double struck upper N 2 ≤ k ∈ N 2≤ k∈ N , any colouring of the edges of upper K Subscript n K n Kₙ with n n n sufficiently large gives a copy of upper C Subscript 2 k C 2 k C₂ₖ which has one of three canonical colour patterns: monochromatic, rainbow or lexicographic. In this paper we show that if p equals omega left parenthesis n Superscript negative 1 plus 1 divided by left parenthesis 2 k minus 1 right parenthesis Baseline log n right parenthesis p = ω ( n − 1 + 1 / ( 2 k − 1 ) log ⁡ n ) p=ω (n-1+1/(2k-1)log n) , then bold upper G left parenthesis n comma p right parenthesis G ( n , p ) G(n,p) will asymptotically almost surely also have the property that any colouring of its edges induces canonical copies of upper C Subscript 2 k